2.72.7 Prove Angle Pair RelationshipsBell Thinger

Give a reason for each statement.

ANSWER Transitive Prop. of Eq.

ANSWER Def. of perpendicular

ANSWER Def. of segment congruence

1. If m 1 = 90º and m 2 = 90º, then m 1 = m 2.

2. If AB BC , then ABC is a right angle.┴

3. If FG RS, then FG = RS=

2.7

2.7Example 1

STATEMENTS REASONS

1.Given1.AB BC , DC BC

2.Definition of perpendicularlines

2. B and C are right angles.

Write a proof.

GIVEN: AB BC , DC BC

PROVE: B C

3.Right Angles CongruenceTheorem

3. B C

2.7

2.7Example 2

Prove that two angles supplementary to the same angle are congruent.

GIVEN: 1 and 2 are supplements.3 and 2 are supplements.

PROVE: 1 3

2.7

STATEMENTS REASONS

Given1.

Example 2

2. m 1+ m 2 = 180°m 3+ m 2 = 180°

2. Definition of supplementary angles

Transitive Property of Equality

3.3. m 1 + m 2 = m 3 + m 2

4. m 1 = m 3 Subtraction Property of Equality

4.

5. 1 3 Definition of congruent angles

5.

1 and 2 are supplements.1.3 and 2 are supplements.

2.7

2.7Example 3

GIVEN: 5 and 7 are vertical angles.

PROVE: 5 7

Prove vertical angles are congruent.

STATEMENTS REASONS

5 and 7 are vertical angles.1. 1. Given

2. 5 and 6 are a linear pair. 6 and 7 are a linear pair.

2. Definition of linear pair, as shown in the diagram

3. 5 and 6 are supplementary. 6 and 7 are supplementary.

3. Linear Pair Postulate

4. 5 7 Congruent Supplements Theorem

4.

2.7Guided Practice2. If m 1 = 112°, find m 2,

m 3, and m 4.

ANSWER m 2 = 68°

m 3 = 112°

m 4 = 68°

3. If m 2 = 67°, find m 1, m 3, and m 4.

ANSWER m 1 = 113°

m 3 = 113°

m 4 = 67°

2.7Guided Practice

4. If m 4 = 71°, find m 1, m 2, and m 3.

ANSWER m 1 = 109°

m 2 = 71°

m 3 = 109°

2.7Example 4

SOLUTION

Because TPQ and QPR form a linear pair, the sum of their measures is 180.

The correct answer is B.

ANSWER

2.7Example 5Tell whether the proof is logically valid. If it is not, explain how to change the proof so that it is valid.

GIVEN: 1 is a right angle.PROVE: 3 is a right angle.

STATEMENTS REASONS

1. 1 is a right angle. 1. Given

3. 3 is a right angle. 3. Right Angles Congruence Theorem

2. 1 3 2. Vertical Angles Congruence Theorem

2.7

The proof is not logically valid. The Right Angles Congruence Theorem does not let you conclude that 3 is a right angle. It just says that all right angles are congruent.

Here is a way to complete the proof.

SOLUTION

Example 5

2.7

REASONSSTATEMENTS

6. 3 is a right angle.

1. 1 is a right angle.

2. 1 3

1. Given

2. Vertical Angles Congruence Theorem

3. Definition of congruent angles

3. m 1 = m 3

4. m 1 = 90º

5. m 3 = 90º

4. Definition of right angle

5. Transitive Property of Equality

6. Definition of right angle

Example 5

2.7Guided Practice5. Solve for x.

x = 49ANSWER

6. Find m TPS.

m TPS = 148°

ANSWER

2.7Exit Slip

1. Give the reason for each step

Def. of linear pair

Given

PROVE : 1 is supplementary to 4

GIVEN : 1 5

Substitution Prop. of Eq.

Def. of supplementary

Linear Pair Post .

Def. of supplementary

STATEMENTS REASONS

2. m 1 = m 5

3. 4 and are a linear pair. 5

1. 1 5

4 and are supplementary .4. 5

m 4 + m 5 = 1805.

m 4 + m 1 = 1806.

7. 1 is supplementary to 4.

Def. of

2.7

Homework

Pg 129-133# 10, 14, 28, 37, 38